ON THE FIRST COHOMOLOGY OF DISCRETE SUBGROUPS OF SEMI-SIMPLE LIE GROUPS.
ON THE FIRST COHOMOLOGY OF DISCRETE SUBGROUPS OF SEMI-SIMPLE LIE GROUPS.
复制标题
关于半单李群离散子群的第一上同调。
DOI：
10.2307/2373227
复制
发表时间：
1965
影响因子：
1.7
通讯作者：
M. Raghunathan
中科院分区：
文献类型：
--
作者：
M. Raghunathan
介绍。令 G 为连通半单李群，r 为离散子群，使得商 G/r 是紧致的。设 po 是 G 的有限维表示。本文的目的是证明，对于一大类表示 po，r 的第一个上同调群，其系数在表示 pr 中（po 对 r 的限制）为零（精确的陈述，请参见定理 1）。我们的结果特别表明，如果 po 不包含 G 的平凡表示，并且如果 G 的简单分量不紧致或局部同构于 SO, (n, 1) 或 SU (n, 1)，则第一个上同调群消失。即使 G 具有与 SOo (n, 1) 或 SU (n, 1) 局部同构的分量，我们也根据 po 的复化 p 的不可约分量的最高权重给出了上同调消失的充分条件。这些上同调群的重要性源于它们在变形理论中所扮演的角色；例如，当 poO 是伴随表示时，这些上同调群与李群离散子群的变形理论密切相关[6]。 A. Weil [8]（另见[5]和[6]）本质上证明了当 po 是伴随表示时，如果 G 没有紧致或三维分量，则该上同调群消失。这个结果是我们定理的一个特例（参见定理 1 的推论 1）。 Y. Matsushima 在[4]中处理了琐碎陈述的情况。
Introduction. Let G be a connected semi-simple Lie group and r a discrete subgroup such that the quotient G/r is compact. Let po be a finite dimensional representation of G. Our aim in this paper, is to show that for a large class of representations po, the first cohomology group of r with coefficients in the representation pr (the restriction of po to r) is zero (for a precise statement, see Theorem 1). Our results say in particular that if po does not contain the trivial representation of G and if no simple component of G is compact or locally isomorphic to SO, (n, 1) or SU (n, 1), then this first cohomology group vanishes. Even if G has components locally isomorphic to SOo (n, 1) or SU (n, 1) we give a sufficient condition for the vanishing of the cohomology in terms of the highest weights of the irreducible components of the complexification p of po. The importance of these cohomology groups arises from the role they play in deformation theory; for instance, when poO is the adjoint representation, these cohomology groups are intimately connected with the theory of deformations of discrete subgroups of Lie groups [6]. It has been proved essentially by A. Weil [8] (see also [5] and [6]) that when po is the adjoint representation, this cohomology group vanishes if G has no compact or three dimensional components. This result is a special case of our theorem (see Corollary 1 to Theorem 1). The case of trivial representations has been treated by Y. Matsushima in [4].